When I move a vertex, it seems like I'm "breaking through" the mesh
I think the issue is with your mirror modifier. You're using the Mirror Modifier to mirror the car's left side to the right side (or vice versa), but since you've dragged some of the edges/vertices across that mirror line, the mirror modifier is duplicating that, creating overlapping surfaces.

Disable the Mirror modifier visibility to see exactly what it's replicating. You'd want to drag all the vertices that cross over the mirror line back. Another way to avoid this type of issue is to enable Clipping in the modifier settings (bottom right of your screenshots). This prevents you from dragging any points across the mirror axis, and will clip any points to that axis.